Abstract

A conflict-free coloring of a hypergraph \(H = (V, \mathcal {E})\) with \(n=|V|\) vertices and \(m=|\mathcal {E}|\) hyperedges (where \(\mathcal {E}\subseteq 2^V\)), is a coloring of the vertices V such that every hyperedge \(E \in \mathcal {E}\) contains a vertex of “unique” color. Our goal is to minimize the total number of distinct colors. In its full generality, this problem is known as the conflict-free (hypergraph) coloring problem. It is known that \(\Theta (\sqrt{m})\) colors might be needed in general.

In this paper we study the relaxation of the problem where one is allowed to assign multiple colors to the same node. The goal here is to substantially reduce the total number of colors, while keeping the number of colors per node as small as possible. By a simple adaptation of a result by Pach and Tardos [2009] on the single-color version of the problem, one obtains that only \(O(\log ^2 m)\) colors in total are sufficient (on every instance) if each node is allowed to use up to \(O(\log m)\) colors.

By improving on the result of Pach and Tardos (under the assumption \(n\ll m\)), we show that the same result can be achieved with \(O(\log m \cdot \log n)\) colors in total, and either \(O(\log m)\) or \(O(\log n\cdot \log \log m) \subseteq O(\log ^2 n)\) colors per node. The latter coloring can be computed by a polynomial-time Las Vegas algorithm.

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